Introducing a Probabilistic Structure on Sequential Dynamical Systems, Simulation and Reduction of Probabilistic Sequential Networks
aa r X i v : . [ q - b i o . GN ] A p r INTRODUCING A PROBABILISTIC STRUCTURE ONSEQUENTIAL DYNAMICAL SYSTEMS, SIMULATION ANDREDUCTION OF PROBABILISTIC SEQUENTIAL NETWORKS
MARIA A. AVINO-DIAZ
Abstract.
A probabilistic structure on sequential dynamical systems is in-troduced here, the new model will be called Probabilistic Sequential Network,PSN. The morphisms of Probabilistic Sequential Networks are defined usingtwo algebraic conditions. It is proved here that two homomorphic ProbabilisticSequential Networks have the same equilibrium or steady state probabilities ifthe morphism is either an epimorphism or a monomorphism. Additionally, theproof of the set of PSN with its morphisms form the category
PSN , havingthe category of sequential dynamical systems
SDS , as a full subcategory isgiven. Several examples of morphisms, subsystems and simulations are given. Introduction
Probabilistic Boolean Networks was introduced by I. Schmulevich, E. Dougherty,and W. Zhang in 2000, for studying the dynamic of a network using time discreteMarkov chains, see [14, 15, 17, 16]. This model had several applications in thestudy of cancer, see [18]. It is important for development an algebraic mathematicaltheory of the model Probabilistic Boolean Network PBN, to describe special mapsbetween two PBN, called homomorphism and projection, the first papers in thisdirection are, [4, 6], ♭ . Instead of this model is being used in applications, theconnection of the graph of genes and the State Space is an interesting problem tostudy. The introduction of probabilities in the definition of Sequential DynamicalSystem has this objective.The theory of sequential dynamical systems (SDS) was born studying networkswhere the entities involved in the problem do not necessarily arrive at a place atthe same time, and it is part of the theory of computer simulation, [2, 3]. Themathematical background for SDS was recently development by Laubenbacher andPareigis, and it solves aspects of the theory and applications, see [8, 9, 10].The introduction of a probabilistic structure on Sequential Dynamical Systemsis an interesting problem that it is introduced in this paper. A SDS induces afinite dynamical system ( k n , f ), [5], but the mean difference between a SDS andFDS is that a SDS has a graph with information giving by the local functions, andan order in the sequential behavior of these local functions. It is known ♭ , thata finite dynamical systems can be studied as a SDS, because we can construct a Key words and phrases. simulation, homomorphism of dynamical systems, sequential dynam-ical systems, probabilistic sequential networks, categories, Markov Chain.
Primary: 05C20, 37B99, 68Q65, 93A30; Secondary:18B20, 37B19, 60G99. ♭ We will use the acronym PBN, PSN, or SDS for plural as well as singular instances. ♭ Information giving by Laubenbacher. bigger system that in this case is sequential. Making together the sequential orderand the probabilistic structure in the dynamic of the system, the possibility to workin applications to genetics increase, because genes act in a sequential manner. Inparticular the notion of morphism in the category of SDS establishes connectionbetween the digraph of genes and the State Space, that is the dynamic of thefunction. Working in the applications, Professor Dougherty’s group wanted toconsider two things in the definition of PBN: a sequential behavior on genes, and theexact definition of projective maps between two PBN that inherits the propertiesof the first digraph of genes. For this reason, a new model that considers bothquestions and tries to construct projections that work well is described here. Iintroduce in this paper the sequential behavior and the probability together inPSN and my final objective is to construct projective maps that let us reducethe number of functions in the finite dynamical systems inside the PBN. One ofthe mean problem in modeling dynamical systems is the computational aspect ofthe number of functions and the computation of steady states in the State Space.In particular, the reduction of number of functions is one of the most importantproblems, because by solving that we can determine which part of the network
StateSpace may be simplified. The concept of morphism, simulation, epimorphism, andequivalent Probabilistic Sequential Networks are developed in this paper, with thisparticular objective.This paper is organized as follows. In section 2, a notation slightly differentto the one used in [9] is introduced for homomorphisms of SDS. This notation ishelpful for giving the concept of morphism of PSN. In section 3, the probabilisticstructure on SDS is introduced using for each vertex of the support graph, a set oflocal functions, more than one schedule, and finally having several update functionswith probabilities assigned to them. So, it is obtained a new concept: probabilisticsequential network (PSN). In Theorem 4.3 is proved that monomorphisms, and epi-morphisms of PSN have the same equilibrium or steady state probabilities . Thesestrong results justify the introduction of the dynamical model PSN as an appli-cation to the study of sequential systems. In section 5, we prove that the PSNwith its morphisms form the category
PSN , having the category
SDS as a fullsubcategory. Several examples of morphisms, subsystems and simulations are givenin Section 6. 2.
Preliminaries
In this introductory section we give the definitions and results of SequentialDynamical System introduced by Laubenbacher and Pareigis in [9]. Let Γ be agraph, and let V Γ = { , . . . , n } be the set of vertices of Γ. Let ( k i | i ∈ V Γ ) be afamily of finite sets. The set k a are called the set of local states at a , for all a ∈ V Γ .Define k n := k × · · · × k n with | k i | < ∞ , the set of (global) states of Γ.A Sequential Dynamical System (SDS) F = (Γ , ( k i ) ni =1 , ( f i ) ni =1 , α )consists of1. A finite graph Γ = ( V Γ , E Γ ) with the set of vertices V Γ = { , . . . , n } , andthe set of edges E Γ ⊆ V Γ × V Γ .2. A family of finite sets ( k i | i ∈ V Γ ). NTRODUCING A PROBABILISTIC STRUCTURE ON SEQUENTIAL DYNAMICAL SYSTEMS3
3. A family of local functions f i : k n → k n , that is f i ( x , . . . , x n ) = ( x , . . . , x i − , f , x i +1 , . . . , x n )where f ( x , . . . , x n ) depends only of those variables which are connected to i in Γ.4. A permutation α = ( α . . . α n ) in the set of vertices V Γ , called anupdate schedule ( i.e. a bijective map α : V Γ → V Γ ).The global update function of the SDS is f = f α ◦ . . . ◦ f α n . The function f defines the dynamical behavior of the SDS and determines a finite directed graphwith vertex set k n and directed edges ( x, f ( x )), for all x ∈ k n , called the StateSpace of F , and denoted by S f .The definition of homomorphism between two SDS uses the fact that the vertices V Γ = { , . . . , n } of a SDS and the states k n together with their evaluation map k n × V Γ ∋ ( x, a ) < x, a > := x a ∈ k i , form a contravariant setup, so that morphismsbetween such structures should be defined contravariantly, i.e. by a pair of certainmaps φ : Γ → ∆ , and the induced function h φ : k m → k n with the graph ∆ having m vertices. Here we use a notation slightly different that the one using in [9].Let F = (Γ , ( f i : k n → k n ) , α ) and G = (∆ , ( g i : k m → k m ) , β ) be two SDS.Let φ : ∆ → Γ be a digraph morphism. Let ( b φ b : k φ ( b ) → k b , ∀ b ∈ ∆) , be afamily of maps in the category of Set . The map h φ is an adjoint map, because isdefined as follows: consider the pairing k n × V Γ ∋ ( x, a ) < x, a > := x a ∈ k a ; andsimilarly k m × V ∆ ∋ ( y, b ) < y, b > := y b ∈ k b . The induced adjoint map holds < h φ ( x ) , b > := ˆ φ b ( < x, φ ( b ) > ) = ˆ φ b ( x φ ( b ) ). Then φ, and ( b φ b ) induce the adjointmap h φ : k n → k m defined as follows:(2.1) h φ ( x , . . . , x n ) = ( b φ ( x φ (1) ) , . . . , b φ m ( x φ ( m ) )) . Then h : F → G is a homomorphism of SDS if for all sets of orders τ β associatedto β in the connected components of ∆, the map h φ holds the following conditions:(2.2) (cid:0) g β l ◦ g β l +1 ◦ · · · ◦ g β s (cid:1) ◦ h φ = h φ ◦ f α i k n f αi −−−−→ k nh φ y h φ y k m g βl ◦···◦ g βs −−−−−−−→ k m where { β l , β l +1 , . . . , β s } = φ − ( α i ). If φ − ( α i ) = ∅ , then Id k m ◦ h φ = h φ ◦ f α i , andthe commutative diagram is now the following:(2.3) k n f αi −−−−→ k nh φ y h φ y k m Id km −−−−→ k m For examples and properties see[9]. It that paper, the authors proved that theabove diagrams implies the following one(2.4) k n f = f α ◦···◦ f αn −−−−−−−−−→ k nh φ y h φ y k m g = g β ◦···◦ g βm −−−−−−−−−→ k m MARIA A. AVINO-DIAZ
Probabilistic Boolean Networks [14, 15, 17, 18] The model Probabilistic BooleanNetwork A = A (Γ , F, C ) is defined by the following:(1) a finite digraph Γ = ( V Γ , E Γ ) with n vertices.(2) a family F = { F , F , . . . , F n } of ordered sets F i = { f i , f i , . . . , f il ( i ) } offunctions f ij : { , } n → { , } , for i = 1 , · · · , n , and j = 1 , . . . , l ( i ) calledpredictors,(3) and a family C = { c ij } i,j , of selection probabilities. The selection proba-bility that the function f ij is used for the vertex i is c ij .The dynamic of the model Probabilistic Boolean Network is given by the vectorfunctions f k = ( f k , f k , . . . , f nk n ) : { , } n → { , } n for 1 ≤ k i ≤ l ( i ), and f ik i ∈ F i , acting as a transition function. Each variable x i ∈ { , } represents thestate of the vertex i . All functions are updated synchronously. At every time step,one of the functions is selected randomly from the set F i according to a predefinedprobability distribution. The selection probability that the predictor f ij is used topredict gene i is equal to c ij = P { f ik i = f ij } = X k i = j p { f = f k } . There are two digraph structures associated with a Probabilistic Boolean Network:the low-level graph Γ, and the high-level graph which consists of the states of thesystem and the transitions between states. The state space S of the network to-gether with the set of network functions, in conjunction with transitions betweenthe states and network functions, determine a Markov chain. The random per-turbation makes the Markov chain ergodic, meaning that it has the possibility ofreaching any state from another state and that it possesses a long-run (steady-state)distribution. As a Genetic Regulatory Network (GRN), evolves in time, it will even-tually enter a fixed state, or a set of states, through which it will continue to cycle.In the first case the state is called a singleton or fixed point attractor, whereas,in the second case it is called a cyclic attractor. The attractors that the networkmay enter depend on the initial state. All initial states that eventually producea given attractor constitute the basin of that attractor. The attractors representthe fixed points of the dynamical system that capture its long-term behavior. Thenumber of transitions needed to return to a given state in an attractor is called thecycle length. Attractors may be used to characterize a cells phenotype (Kauffman,1993) [7]. The attractors of a Probabilistic Genetic Regulatory Network (PGRN)are the attractors of its constituent GRN. However, because a PGRN constitutesan ergodic Markov chain, its steady-state distribution plays a key role. Dependingon the structure of a PGRN, its attractors may contain most of the steady-stateprobability mass [1, 12, 19].3. Probabilistic Sequential Networks
The following definition give us the possibility to have several update functionsacting in a sequential manner with assigned probabilities. All these, permit usto study the dynamic of these systems using Markov chains and other probabilitytools.
Definition 3.1.
A Probabilistic Sequential Network (PSN) D = (Γ , { F a } | Γ | = na =1 , ( k a ) na =1 , ( α j ) mj =1 , C = { c , . . . , c s } ) NTRODUCING A PROBABILISTIC STRUCTURE ON SEQUENTIAL DYNAMICAL SYSTEMS5 consists of: (1) a finite graph
Γ = ( V Γ , E Γ ) with n vertices; (2) a family of finite sets ( k a | a ∈ V Γ ) . (3) for each vertex a of Γ a set of local functions F a = { f ai : k n → k n | ≤ i ≤ ℓ ( i ) } , is assigned. (i. e. there exists a bijection map ∼ : V Γ → { F a | ≤ a ≤ n } )(for definition of local function see ( ?? .2)). (4) a family of m permutations α = ( α . . . α n ) in the set of vertices V Γ . (5) and a set C = { c , . . . , c s } , of assign probabilities to s update functions. We select one function in each set F a , that is one for each vertices a of Γ, anda permutation α , with the order in which the vertex a is selected, so there are n possibly different update functions f i = f α i ◦ . . . ◦ f α n i n , where n ≤ n ! × ℓ (1) ×· · · × ℓ ( n ). The probabilities are assigned to the update functions, so there exists aset S = { f , . . . , f s } of selected update functions such that c i = p ( f i ), 1 ≤ i ≤ s . Definition 3.2.
The State Space of D is a weighted digraph whose vertices are theelements of k n and there is an arrow going from the vertex u to the vertex v if thereexists an update function f i ∈ S , such that v = f i ( u ) . The probability p ( u, v ) of thearrow going from u to v is the sum of the probabilities c f i of all functions f i , suchthat v = f i ( u ) , u p ( u,v ) −→ f i ( u ) = v . We denote the State Space by S D . For each one update function in S we have one SDS inside the PSN, so the StateSpace S f is a subdigraph of S D . When we take the whole set of update functionsgenerated by the data, we will say that we have the full PSN. It is very clear thata SDS is a particular PSN, where we take one local function for each vertex, andone permutation. The dynamic of a PSN is described by Markov Chains of thetransition matrix associated to the State Space.
Example 3.3.
Let D = (Γ; F , F , F ; Z ; α , α ; ( c f i ) i =1 ) , be the following PSN: (1) The graph Γ : • • (cid:31) | • . (2) Let x = ( x , x , x ) ∈ { , } . In this paper, we always consider the operationsover the finite field Z = { , } , but we use additionally the following notation ¯ x = x + 1 . Then the sets of local functions from Z → Z are the following F = { f ( x ) = (1 , x , x ) , f ( x ) = ( ¯ x , x , x )) } F = { f ( x ) = ( x , x x , x ) } F = { f ( x ) = ( x , x , x x ) , f ( x ) = ( x , x , x x + x ) } . (3) The schedules or permutations are α = (cid:0) (cid:1) ; α = (cid:0) (cid:1) . Weobtain the following table of functions, and we select all of them for D because theprobabilities given by C . f = f ◦ f ◦ f f = f ◦ f ◦ f f = f ◦ f ◦ f f = f ◦ f ◦ f f = f ◦ f ◦ f f = f ◦ f ◦ f f = f ◦ f ◦ f f = f ◦ f ◦ f . MARIA A. AVINO-DIAZ
The update functions are the following: f ( x ) = (1 , x , x ) f ( x ) = (1 , x x , x x ) f ( x ) = (1 , x , x + x ) f ( x ) = (1 , x x , x x + x ) f ( x ) = ( ¯ x , ¯ x x , ( x + 1) x ) f ( x ) = ( ¯ x , x x , x x ) f ( x ) = ( ¯ x , ( x + 1) x , ( x + 1) x + x ) f ( x ) = ( ¯ x , x x , x x + x ) . (4) The probabilities assigned are the following: c f = . c f = . c f = . c f = . c f = . c f = . c f = . c f = . . Example 3.4.
We notice that there are several PSN that we can construct withthe same initial data of functions and permutations, but with different set of prob-abilities, that is, subsystems of D . For example if S ′ = { f , f , f , f } , F ′ = { f } ,and D = { d f = . , d f = . , d f = . , d f = . } , then B = (Γ; F ′ , F , F ; Z ; α , α ; D = { . , . , . , . } ) , is a PSN too. Morphisms of Probabilistic Sequential Networks
The definition of morphism of PSN is a natural extension of the concept ofhomomorphism of SDS. In this section we prove in Theorem 4.2 a strong property,that is the distribution of probabilities of two homomorphic PSN are enough closeto prove Theorem 4.3.Consider the following two PSN D = (Γ , ( F a ) | Γ | = na =1 , ( k a ) na =1 , ( α j ) j , C ) and D = (∆ , ( G b ) | ∆ | = mb =1 , ( k b ) mb =1 , ( β j ) j , D ) . We denote by S i the set of update functionsof D i , i = 1 ,
2; and the following notation for ( u, v ) ∈ k n × k n , and ( w, z ) ∈ k m × k m , c f ( u, v ) = (cid:26) p ( f ) if f ( u ) = v otherwise (cid:27) , d g ( w, z )) = (cid:26) p ( g ) if g ( w ) = z otherwise (cid:27) where p ( h ) is the probability of the function h . Definition 4.1. (Homomorphisms of PSN)
A morphism h : D → D consist of: (1) A graph morphism φ : ∆ → Γ , and a family of maps in the category Set , ( b φ b : k φ ( b ) → k b ∀ b ∈ ∆) , that induces the adjoint function h φ , see (2.1). (2) The induced adjoint map h φ : k n → k m holds that for all update functions f in S there exists an update function g ∈ S such that h is a SDS-morphism from (Γ , ( f : k n → k n ) , α j ) to (∆ , ( g : k m → k m ) , β j ) . That is,the diagrams 2.2, 2.3, and 2.4 commute for all f and its selected g . (4.1) k n f = f α ◦···◦ f αn −−−−−−−−−→ k nh φ y h φ y k m g = g β ◦···◦ g βm −−−−−−−−−→ k m The second condition induces a map µ from S to S , that is µ ( f ) = g if theselected function for f is g . We say that a morphism h from D to D is a PSN-isomorphism if φ , h φ , and µ are bijective functions, and d ( h φ ( u ) , h φ ( g ( u )) = c ( u, f ( u )) for all u , in k n , and all f ∈ S , and all g ∈ S . We denote it by D ∼ = D . Special morphisms.
Let D = (Γ , ( F i ) ni =1 , ( α j ) j ∈ J , C ) be a PSN. NTRODUCING A PROBABILISTIC STRUCTURE ON SEQUENTIAL DYNAMICAL SYSTEMS7
Identity morphism.
The functions φ = id Γ , h φ = id k n , and µ = id S define the identity morphism I : D → D , and it is a trivial example of a PSN-isomorphism.
Monomorphism
A morphism h of PSN is a monomorphism if φ is surjective, h φ is injective, and for all f , and its associated g we have that d g ≤ c f . Epimorphism
A morphism is an epimorphism if φ is injective, h φ is surjective,and for all f , and its associated g we have that d g ≥ c f . Remark
If the morphism h is either a monomorphism or an epimorphism, thenthe function µ is not necessary injective, neither surjective. Some theorems
Theorem 4.2.
The morphism h : D → D induces the following probabilisticcondition:For a fixed real number ≤ ǫ < , the map h φ satisfies the following: (4.2) max u,v | c f ( u, v ) − d g ( h φ ( u ) , h φ ( v )) | ≤ ǫ for all f in S , and its selected g in S , and all ( u, v ) ∈ k n × k n .Proof. Suppose φ, and h φ satisfy the Definition 4.1; and | c f ( u, v ) − d g ( h φ ( u ) , h φ ( v )) | ≥ u, v ) ∈ k n × k n . Then we have one of the following cases1. c f ( u, v ) = 1 and d g ( h φ ( u ) , h φ ( v )) = 0. It is impossible by condition (2) indefinition 4.1. In fact, if we have an arrow going from u to v = f ( u ), thenthere exists an arrow going from h φ ( u ) to h φ ( v ) = g ( h φ ( u )) by diagram4.1, and the probability d g ( h φ ( u ) , h φ ( v )) = 0.2. c f ( u, v ) = 0, and d g ( h φ ( u ) , h φ ( v )) = 1. It is impossible because at leastthere exists one element v ∈ k n , such that f ( u ) = v ∈ k n and c f ( u, v ) =0, then d g ( h φ ( u ) , h φ ( v )) = 0 too. Since the sum of probabilities of allarrow going up from h φ ( u ) is equal 1, then d g ( h φ ( u ) , h φ ( v )) <
1, and ourclaim holds.Therefore the condition holds, and always ǫ exists. (cid:3) In the next theorem we will use the following notation:(1) S φ = µ ( S ).(2) g t = g ◦ g ◦ · · · ◦ g , t times.(3) p t ( u, v ) = P f t c f t ( u, v ), and p t ( h φ ( u ) , h φ ( v )) = P g t d g t ( h φ ( u ) , h φ ( v ))(4) T i denotes the transition matrix of D i , for i = 1 ,
2, and p t ( u, v ) = ( T it ) ( u,v ) . Theorem 4.3. If h : D −→ D is either a monomorphism or an epimorphism ofprobabilistic sequential networks, then: lim t →∞ | ( T ) tu,v − ( T ) th φ ( u ) ,h φ ( v ) | = 0 , for all ( u, v ) ∈ k n × k n . That is, the equilibrium state of both systems are equals.Proof. The condition giving by Theorem 4.2 asserts that, there exists a fixed realnumber 0 ≤ ǫ <
1, such that the map h φ satisfies the following:max u,v | c f ( u, v ) − d g ( φ ( u ) , φ ( v )) | ≤ ǫ for all f in S , and its selected g in S , and all ( u, v ) ∈ k n × k n .If there is a function f going from u to v = f ( u ) in k n , then there exists afunction g going from h φ ( u ) to h φ ( v ), such that g ( h φ ( u )) = h φ ( f ( u )). MARIA A. AVINO-DIAZ
Because c f ( u, f ( u )) = c f ( u, f ( u )) c f ( f ( u ) , f ( u )) = c f , and d g ( h φ ( u ) , g ( h φ ( u ))) = d g ( h φ ( u ) , g ( h φ ( u ))) d g ( g ( h φ ( u )) , g ( h φ ( u ))) = d g . We have | c f ( u, f ( u )) − d g ( h φ ( u ) , g ( h φ ( u ))) | = | c f − d g | If h is a monomorphism, then c f ≥ d g , for all f and its associated g . Then | c f ( u, f ( u )) − d g ( h φ ( u ) , g ( h φ ( u ))) | = | c f − d g | ≤ c f . By induction we have that | c f t ( u, f t ( u )) − d g t ( h φ ( u ) , g t ( h φ ( u ))) | = | c f t − d gt | ≤ c f t . This result implies that | p t ( u, v ) − p t ( h φ ( u ) , h φ ( v )) | ≤ X f t c f t + δ t ( u, v )where δ t ( u, v ) = P g ∈ G ( u,v ) d gt , and G ( u, v ) = { g ∈ G | g ( h φ ( u )) = h φ ( v ) , and g = µ ( S ) } .Then, when t goes to infinity the sum P f t c f t + δ t ( u, v ) goes to 0, and thetheorem holds. If h is an epimorphism we obtain the same results, so the theoremholds again. (cid:3) The category
PSN
In this section, we prove that the PSN with the morphisms form a category, thatwe denote by
PSN . For definitions, and results in Categories see [11].
Theorem 5.1.
Let h = ( φ , h φ ) : D → D and h = ( φ , h φ ) : D → D betwo morphisms of PSN. Then the composition h = ( φ, h φ ) = ( φ , h φ ) ◦ ( φ , h φ ) = h ◦ h : D → D is defined as follows: h = ( φ, h φ ) = ( φ ◦ φ , h φ ◦ h φ ) is amorphism of PSN. The function µ h = µ h ◦ µ h . Proof.
The composite function φ = φ ◦ φ of two graph morphisms is again a graphmorphism. The composite function h φ = h φ ◦ h φ is again a digraph morphismwhich satisfies the conditions in Definition 4.1, by Proposition and Definition 2.7in [9]. So, h = ( φ, h φ ) is again a morphism. of PSN. (cid:3) Theorem 5.2.
The Probability Sequential Networks together with the homomor-phisms of PSN form the category
PSN .Proof.
Easily follows from Theorem 5.1. (cid:3)
Theorem 5.3.
The SDS together with the morphisms defined in [9] form a fullsubcategory of the category
PSN .Proof.
It is trivial. (cid:3)
NTRODUCING A PROBABILISTIC STRUCTURE ON SEQUENTIAL DYNAMICAL SYSTEMS9 Simulation and examples
In this section we give several examples of morphisms, and simulations. In thesecond example we show how the Definition 4.1 is verified under the suppositionthat a function φ is defined. So, we have two examples in (6.2), one with φ thenatural inclusion, and the second with φ a surjective map. The third, and thefourth examples are morphisms that represent simulation of G by F . We begin thissection with the definitions of Simulation and sub-PSN. Definition of Simulation in the category
PSN . The probabilistic sequen-tial network G is simulated by F if there exists a monomorphism h : F → G or anepimorphism h ′ : G → F . Sub Probabilistic Sequential Network
We say that a PSN G is a sub Prob-abilistic Sequential Network of F if there exists a monomorphism from G to F . Ifthe map µ is not a bijection, then we say that it is a proper sub-PSN.6.1. Examples.(6.1.1)
In the examples 3.3, and 3.4 we define two PSN D and B . The functions φ = Id Γ , h φ = Id k n , and µ the natural inclusion from S to S define the inclusion ι µ : B → D . It is clear that the inclusion is a monomorphism, so D is simulated by B . (6.1.2) Consider the two graphs belowΓ 2 • • | | • • • • | • Suppose that the functions associated to the vertices are the families { f , f , f , f } ,for Γ and { g , g , g } for ∆. The permutations are α = (4 3 2 1), α = (4 1 3 2)and β = (3 2 1), β = (1 3 2), so, S = { f = f ◦ f ◦ f ◦ f ; f = f ◦ f ◦ f ◦ f } ,and X = { g = g ◦ g ◦ g ; g = g ◦ g ◦ g } . Then, we have constructed two PSN,each one with two permutations and only one function associated to each vertex inthe graph; denoted by: D = (Γ; f , f , f , f ; α , α ; C ) and B = (∆; g , g , g ; β , β ; D ) . Case (a)
We assume that there exists a morphism h : D → B , with the graphmorphism φ : ∆ → Γ giving by φ (1) = 1 , φ (2) = 2 , φ (3) = 3. Suppose thefunctions ( b φ b : k φ ( b ) → k b , ∀ b ∈ ∆) , are giving, and the adjoint function h φ : k → k , h φ ( x , x , x , x ) = ( ˆ φ ( x ) , ˆ φ ( x ) , ˆ φ ( x ))is defined too. If h is a morphism, which satisfies the definition (4.1), then thefollowing diagrams commute: k f −−−−→ k f −−−−→ k f −−−−→ k f −−−−→ k h φ y h φ y h φ y h φ y h φ y k Id −−−−→ k g −−−−→ k g −−−−→ k g −−−−→ k , k f −−−−→ k f −−−−→ k f −−−−→ k f −−−−→ k h φ y h φ y h φ y h φ y h φ y k g −−−−→ k Id −−−−→ k g −−−−→ k g −−−−→ k ,k f −−−−→ k h φ y h φ y k g −−−−→ k k f −−−−→ k h φ y h φ y k g −−−−→ k . Case (b)
Consider now the map φ : Γ → ∆, defined by φ (1) = 1, φ (2) = 2, φ (3) = 3, and φ (4) = 3. If there exists a morphism h : B → D that satisfiesDefinition 4.1, then k g −−−−→ k g −−−−→ k g −−−−→ k h φ y h φ y h φ y h φ y k f ◦ f −−−−→ k f −−−−→ k f −−−−→ k ,k g −−−−→ k g −−−−→ k g −−−−→ k h φ y h φ y h φ y h φ y k f −−−−→ k f ◦ f −−−−→ k f −−−−→ k , (6.1.3) We now construct a monomorphism h : F → G , with the properties that φ is surjective and the function h φ is injective. The PSN F = (Γ , ( F i ) , β, C ) hasthe support graph Γ with three vertices, and the PSN G = (∆ , ( G i ) , α, D ) has thesupport graph ∆ with four verticesΓ • | • • • • (cid:31) | • • h : F → G , has the contravariant graph morphism φ : ∆ → Γ,defined by the arrows of graphs, as follows φ (1) = 1, φ (2) = φ (3) = 2 , and φ (4) = 3,so it is a surjective map. The family of functions ˆ φ i : k φ ( i ) → k ( i ) , ˆ φ ( x ) = x ;ˆ φ ( x ) = x ; ˆ φ ( x ) = x ; ˆ φ ( x ) = x , are injective functions. The sets k a = Z ,for all vertices a in ∆, and Γ. The adjoint function is h φ : Z → Z , defined by h φ ( x , x , x ) = ( ˆ φ ( x ) , ˆ φ ( x ) , ˆ φ ( x ) , ˆ φ ( x )) = ( x , x , x , x ) . Then, the first condition in the definition 4.1 holds.The PSN F = ( Γ; ( F i ) ; β ; C ) , is defined with the following data.The set of functions F = { f , f ) } , F = { f } , and F = { f } , where f = Id, f ( x , x , x ) = (1 , x , x ) , f = Id,f ( x , x , x ) = ( x , x , x x ) . A permutation β = ( 1 2 3 ); and the probabilities C = { c f = . , c f = . } .So, we are taking all the update functions S = { f , f } ; f = f ◦ f ◦ f , f = ( x , x , x ) = ( x , x , x x );and f = f ◦ f ◦ f , f ( x , x , x ) = (1 , x , x x ) . NTRODUCING A PROBABILISTIC STRUCTURE ON SEQUENTIAL DYNAMICAL SYSTEMS11
On the other hand, the PSN G = (∆; ( G i ) ; α ; D ) has the following data.The families of functions: G = { g , g } ; G = { g , g } , G = { g , g } ; and G = { g } , where g ( x , x , x , x ) = (1 , x , x , x ) g ( x , x , x , x ) = ( x , , x , x ) g ( x , x , x , x ) = ( x , x , x x , x ) g ( x , x , x , x ) = ( x , x , x , x x ) g = Id = g g ( x , x , x , x ) = ( x , x , x , x ) . One permutation or schedule α = (cid:0) (cid:1) . The assigned probabilities d g = . , d g = . , d g = . , d g = . X = { g , g , g , g } : therefore the all update functions are thefollowing g = g ◦ g ◦ g ◦ g , g = g ◦ g ◦ g ◦ g g = g ◦ g ◦ g ◦ g ,g = g ◦ g ◦ g ◦ g g = g ◦ g ◦ g ◦ g , g = g ◦ g ◦ g ◦ g g = g ◦ g ◦ g ◦ g , g = g ◦ g ◦ g ◦ g . The selected functions are g ( x , x , x , x ) = ( x , x , x x , x x ) , g ( x , x , x , x ) = (1 , x , x x , x x ) g ( x , x , x , x ) = ( x , x , x , x x ) , g ( x , x , x , x ) = (1 , x , x , x x ) . We claim that h : F → G is a morphism, in fact the following diagrams commute. Z f −−−−→ Z h φ y h φ y Z g −−−−→ Z , and Z f −−−−→ Z h φ y h φ y Z g −−−−→ Z . In fact, ( h φ ◦ f )( x , x , x ) = ( x , x , x , x x ) = ( g ◦ h φ )( x , x , x ) , on theother hand ( h φ ◦ f )( x , x , x ) = (1 , x , x , x x ) = ( g ◦ h φ )( x , x , x ) so, theproperty holds. We verify the second property in the definition of morphism forthe compositions f and g , and also with the compositions f and g . That is, wecheck the sequence of local functions too. Z f −−−−→ Z f −−−−→ Z f −−−−→ Z h φ y h φ y h φ y h φ y Z g −−−−→ Z g ◦ g −−−−−→ Z g −−−−→ Z Z f −−−−→ Z f −−−−→ Z f −−−−→ Z h φ y h φ y h φ y h φ y Z g −−−−→ Z g ◦ g −−−−−→ Z g −−−−→ Z ( h φ ◦ f )( x , x , x ) = ( x , x , x , x x ) = ( g ◦ h φ )( x , x , x ) , ( h φ ◦ f )( x , x , x ) = ( x , x , x , x ) = (( g ◦ g ) ◦ h φ )( x , x , x ) , ( h φ ◦ f )( x , x , x ) = ( x , x , x , x ) = ( g ◦ h φ )( x , x , x ) , ( h φ ◦ f )( x , x , x ) = (1 , x , x , x ) = ( g ◦ h φ )( x , x , x ) .The probabilities satisfy the following conditions: p ( f ) ≥ p ( g ), and p ( f ) ≥ p ( g ).Then our claim holds, and h φ is a monomorphism. (6.1.4) We can construct an epimorphism h ′ : G → F , that is, the function φ isinjective and the function h ′ φ is surjective. We use φ ′ : Γ → ∆, defined as follow φ ′ ( i ) = i + 1, for all i ∈ V Γ . Therefore ˆ φ ′ i : k φ ′ ( i ) → k ( i ) , ˆ φ ′ i : Z → Z , for all i ∈ V Γ , and should be satisfies < h φ ( x ) , i > := ˆ φ b ( < x, φ ( i ) > ) = ˆ φ b ( x φ ( i ) ). So, theadjoint function is h ′ φ ( x , x , x , x ) = ( ˆ φ ′ ( x ) , ˆ φ ′ ( x ) , ˆ φ ′ ( x )) = ( x , x , x )andsatisfies the following commutative diagrams Z g −−−−→ Z h ′ φ y h ′ φ y Z f −−−−→ Z , Z g −−−−→ Z h ′ φ y h ′ φ y Z f −−−−→ Z , Z g −−−−→ Z h ′ φ y h ′ φ y Z f −−−−→ Z and Z g −−−−→ Z h ′ φ y h ′ φ y Z f −−−−→ Z . These implies that µ ( g ) = µ ( g ) = f , and µ ( g ) = µ ( g ) = f . In fact,( h ′ φ ◦ g )( x , x , x , x ) = ( x , x , x x ) = ( f ◦ h ′ φ )( x , x , x , x ) , ( h ′ φ ◦ g )( x , x , x , x ) = (1 , x , x ¯ x ) = ( f ◦ h ′ φ )( x , x , x , x ) , ( h ′ φ ◦ g )( x , x , x , x ) = ( x , x , x x ) = ( f ◦ h ′ φ )( x , x , x , x ) , ( h ′ φ ◦ g )( x , x , x , x ) = (1 , x , x ¯ x ) = ( f ◦ h ′ φ )( x , x , x , x ) . Checking the compositions of local functions g = g ◦ g ◦ g ◦ g , and f = f ◦ f ◦ f , we have that the following diagrams commute Z g −−−−→ Z g −−−−→ Z g −−−−→ Z g −−−−→ Z h ′ φ y h ′ φ y h ′ φ y h ′ φ y h ′ φ y Z f −−−−→ Z Id −−−−→ Z f −−−−→ Z f −−−−→ Z . By the data we only need to check the following compositions h ′ φ ( g ( x , x , x , x )) = ( x , x , x ) = f ( h ′ φ ( x , x , x , x )) ,h ′ φ ( g ( x , x , x , x )) = ( x , x , x ¯ x ) = f ( h ′ φ ( x , x , x , x )) . Similarly, we canprove that the other functions hold the property. The probabilities satisfy thefollowing conditions: p ( g ) ≤ p ( f ), p ( g ) ≤ p ( f ), p ( g ) ≤ p ( f ), and p ( g ) ≤ p ( f ).Therefore h ′ φ is an epimorphism.7. Equivalent Probabilistic Sequential Networks
Definition 7.1. (Equivalent PSN)
If the morphism h : D → D satisfies that φ , h φ and µ are bijective functions, but the probabilities are not necessary equals, wesay that D , and D are equivalent PSN. We write D ≃ D . So, D , and D are equivalents if there exist ( φ, h φ , µ ), and ( φ − , h − φ , µ − ), suchthat for all update functions f ∈ D and its selected function g ∈ D , the condition f = h − φ ◦ g ◦ h φ holds . It is clear that this relation is an equivalence relation inthe set of PSN. Proposition 7.2. If D ≃ D , then the transition matrices T and T satisfy: ( T m ) ( u,v ) = 0 , if and only if ( T m ) ( h φ ( u ) ,h φ ( v )) = 0 , for all m ∈ N , ( u, v ) ∈ k n × k n .Proof. It is obvious. (cid:3)
Acknowledgment
This work was supported by the Partnership M. D. Anderson Cancer Center,University of Texas, and the Medical Sciences Cancer Center, University of PuertoRico, by the Program AABRE of Rio Piedras Campus, and by the SCORE Programof NIH, Rio Piedras Campus.
NTRODUCING A PROBABILISTIC STRUCTURE ON SEQUENTIAL DYNAMICAL SYSTEMS13
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Gauss Laboratory, Univiversity of Puerto Rico, Rio Piedras, Puerto Rico
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